UPSC NDA I 2015 Mathematics Solved Paper

A function f(x) has inverse function in its range if it is an increasing function in its entire domain
f(x)=x3
⇒f′(x)=3x2
f′(x)≥0‌∀x∊R
Hence x3 is an increasing function in its domain.
So it will have inverse defined in its range
f(x)=sinx⇒f′(x)=cosx
cosx is positive in (0,

π

2

) and negative in (

π

2

,

3π

2

)
So sinx is not an increasing function in the given domain.So inverse of sinx is not defined on its range for the given domain.
f(x)=ex⇒f′(x)=ex,which is greater than 0
So f(x) is an increasing function in its domain.
So inverse of f(x) is defined on its range for the given domain.
Hence, inverse of function in (1) and (3) are defined in their ranges.